A finiteness result on representations of Nori's fundamental group scheme

TL;DR

The paper proves that for a pointed geometrically connected smooth projective variety $(X,x)$ over a sub-$p$-adic field $K$ and for any fixed rank $n$, there are only finitely many isomorphism classes of representations $\pi_{1}^{EF}(X,x) \to GL_n$, equivalently finitely many essentially finite bundles of rank $n$. The proof diverges from Gasbarri's earlier approaches and relies on Jordan's theorem to bound finite subgroups of $GL_n$ and arithmetic properties over $p$-adic fields, together with a controlled reduction to finite torsors via a finite étale cover and careful abelianization arguments. A key outcome is the construction of a universal bound $J(K,n,(X,x))$ governing the size of abelian quotients, enabling a finite enumeration of possible torsors and corresponding representations. Overall, the result establishes non-abelian finiteness for Nori's fundamental group in this setting, contributing to understanding of the arithmetic constraints on representations of $\pi_{1}^{EF}(X,x)$ and the moduli of essentially finite bundles.

Abstract

Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. For any given rank $n$, we prove that there are only finitely many isomorphism classes of representations $π_{1}^{EF}(X,x)\rightarrow \mathrm{GL}_{n}$, where $π_{1}^{EF}(X,x)$ is Nori's fundamental group of essentially finite bundles. Equivalently, there are only finitely many isomorphism classes of essentially finite bundles of rank $n$. This answers a question from C.Gasbarri.

Theorems & Definitions (17)

• Theorem 1.1: Theorem \ref{['main_thm1']}
• Lemma 2.1
• proof
• Lemma 2.2: Proposition 3.5 10.1215/S0012-7094-03-11723-8
• proof
• Lemma 2.3
• proof
• Proposition 2.4: Proposition 14 II §5 lang1994algebraic
• Theorem 2.5: jordan1878memoire
• Corollary 2.6